Theorem lcdsbase

Description: Base set of scalar ring for the closed kernel dual of a vector space. (Contributed by NM, 18-Mar-2015)

	Ref	Expression
Hypotheses	lcdsbase.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcdsbase.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcdsbase.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
	lcdsbase.l	⊢ 𝐿 = ( Base ‘ 𝐹 )
	lcdsbase.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	lcdsbase.s	⊢ 𝑆 = ( Scalar ‘ 𝐶 )
	lcdsbase.r	⊢ 𝑅 = ( Base ‘ 𝑆 )
	lcdsbase.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion	lcdsbase	⊢ ( 𝜑 → 𝑅 = 𝐿 )

Proof

Step	Hyp	Ref	Expression
1		lcdsbase.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcdsbase.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		lcdsbase.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
4		lcdsbase.l	⊢ 𝐿 = ( Base ‘ 𝐹 )
5		lcdsbase.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
6		lcdsbase.s	⊢ 𝑆 = ( Scalar ‘ 𝐶 )
7		lcdsbase.r	⊢ 𝑅 = ( Base ‘ 𝑆 )
8		lcdsbase.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		eqid	⊢ ( oppr ‘ 𝐹 ) = ( oppr ‘ 𝐹 )
10	1 2 3 9 5 6 8	lcdsca	⊢ ( 𝜑 → 𝑆 = ( oppr ‘ 𝐹 ) )
11	10	fveq2d	⊢ ( 𝜑 → ( Base ‘ 𝑆 ) = ( Base ‘ ( oppr ‘ 𝐹 ) ) )
12	9 4	opprbas	⊢ 𝐿 = ( Base ‘ ( oppr ‘ 𝐹 ) )
13	11 7 12	3eqtr4g	⊢ ( 𝜑 → 𝑅 = 𝐿 )